Evidence of chaos in the Belief Propagation for LDPC codes

In this paper, we investigate the behaviors of the Belief Propagation algorithm considered as a dynamic system. In the context of LDPC (Low Density Parity-Check) codes, we use the noise power of the transmission channel as a potentiometer to evaluate the different motions that the BP can follow. The computations of dynamic quantifiers as the bifurcation diagram, the Lyapunov exponent and the reconstructed trajectory enable to bring out four main behaviors. In addition, we propose a novel measure that is the hyperspheres method, which provides the knowledge of the time evolution of the attractor size. The information collected from these different quantifiers helps to better understand the BP evolution and to focus on the noise power values for which the BP suffers from chaos.